Higher-order topological insulators and superconductors are topological phases that exhibit novel boundary states on corners or hinges. Recent experimental advances in controlling dissipation such as gain and loss in atomic and optical systems provide a powerful tool for exploring non-Hermitian topological phases. Here we show that higher-order topological corner states can emerge by introducing staggered on-site gain and loss to a Hermitian system in a trivial phase. For such a non-Hermitian system, we establish a general bulk-corner correspondence by developing a biorthogonal nested-Wilson-loop and edge-polarization theory, which can be applied to a wide class of non-Hermitian systems with higher-order topological orders. The theory gives rise to topological invariants characterizing the non-Hermitian topological multipole moments (i.e., corner states) that are protected by reflection or chiral symmetry. Such gain- and loss-induced higher-order topological corner states can be experimentally realized using photons in coupled cavities or cold atoms in optical lattices.